Exterior billiards: systems with impacts outside bounded domains

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systems with impacts outside bounded domains

Alexander Plakhov

Department of Mathematics, University of Aveiro, Portugal

and

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Abstract

The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics.

We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems.

First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist ’almost perfectly streamlined’ bodies.

Next we consider the generalization of Newton’s problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends.

Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface.

Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direc-tion of the disc modirec-tion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transver-sal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation.

Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retrore-flectors: bodies with specular surface that reverse the direction of any incident beam of light.

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MSC:37D50, 49Q10, 49Q20, 49K30, 37N05, 76G25, 76M28

Keywords: billiards, scattering by obstacles, Newton’s aerodynamic problem, bodies of maximal and minimal resistance, optimal mass transportation, shape optimization, invisible bodies, retroreflectors, rough surface, Magnus effect, free-molecular flow.

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Preface

Imagine that we are going to design a spaceship for a long voyage in open space. During the voyage the ship will cross huge rarefied clouds of interstellar gas. Our goal is to make its shape as streamlining as possible, so that the velocity loss when moving in the clouds is minimal.

In order to specify this task, we need to make a number of assumptions concerning the state of the cloud, its interaction with the spaceship surface, the kind of the ship motion, as well as description of admissible shapes the ship can take (in what follows the spaceship will be called the body, and the cloud, the medium). It is always assumed in this book that the medium is homogeneous and consists of point particles, besides the following conditions are fulfilled:

1) the particles of the medium do not interact with each other;

2) when hitting the body surface, the particles are reflected in the perfectly elastic manner.

The condition 1 is ensured by the fact that the space cloud is highly rarefied, so that mu-tual interaction of particles can be neglected. The condition 2 means that the interaction of particles with the body is billiard-like.

Different settings of the problem correspond to the cases where the medium temper-ature equals zero and where it is positive. The zero-tempertemper-ature assumption is justified in the case where the velocity of thermal motion of the particles is much smaller than the spaceship velocity, and usually significantly simplifies the task. Further, the problem settings and methods of study are completely different in the case of translational motion and in the case where the body performs both translational and rotational motion. Fi-nally, the kind of the problem and approaches to its solution vary greatly depending on the class of admissible bodies.

In particular, in the case of translational motion of convex bodies the drag force (usually called the resistance) can be represented analytically as a functional of the body shape, and variational methods can be used to solve the minimum resistance problem. This kind of problem has a long history originating from the publication by I. Newton in his Principia of the famous problem on minimal resistance of convex axisymmetric bodies and continuing nowadays in a series of paper in 1990s and 2000s related to minimal resistance of convex (not necessarily symmetric) bodies [14, 13, 9, 35, 34]. If we consider nonconvex bodies, an explicit analytical expression for the resistance becomes impossible and one needs to use billiard techniques to minimize the resistance. If, additionally, the body rotates in the course of forward motion, one has to appeal to methods of optimal mass transportation.

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In the book a review of these problems is given and methods of their solution are described. The main part of the book is dedicated to results obtained by the author and his collaborators. The most attention is given to the case where the body is nonconvex and therefore reflections of the particles from its surface are generally multiple. The chapter 3 related to the motion of convex bodies in media with nonzero temperature is an exception.

These problems originating from classical mechanics also allow a natural interpretation from the viewpoint of geometrical optics where the particles incident on the body are replaced with light rays falling on the specular surface of the body and reflecting according to the rule ’the angle of incidence equals the angle of reflection’. In some cases the optical setting is in better agreement with the empirical reality than the mechanical one. Indeed, light rays practically do not mutually interact, and the elastic reflection law approximation is usually much more precise for them than for gas particles.

The optical problems on light scattering by a reflecting surface have their own specific character. We consider, in particular, problems on invisible bodies and retroreflectors. Invisibility in a certain direction means that any light ray falling on the body in this direction and its extension behind the point of last reflection lie on the same straight line. A retroreflector is a body that changes the direction of any incident light ray to the opposite. A well known example of ’partial’ retroreflector is the inner part of a cube corner: a portion of incident light rays make 3 successive reflections from its faces and then move in the direction opposite to the direction of incidence. From the mechanical point of view, an invisible body has zero resistance when moving through a medium in a fixed direction, and a retroreflector has the greatest possible resistance when moving in any direction.

The next important problem is related to description of elastic scattering of particles by a rough surface. We consider a surface that looks smooth for a ’naked eye’, but contains ’microscopic’ unevenness invisible for the eye: dimples, grooves, cracks, etc. A point particle falling on the body and going into a dimple or groove makes one or several reflections there and eventually escapes in a direction that does not obey the law ’the angle of incidence equals the angle of reflection’. Moreover, one cannot predict the direction of escape; instead, the statistical distribution for this direction can be determined. That is, the billiard scattering law at a given point of the surface and for a given velocity of incidence should describe the probability distribution over the velocities of escape. We will see below that it is natural to define the scattering law at a point as a joint distribution of the pair of vectors (velocity of incidence, velocity of reflection), and the law of scattering by a whole rough surface is naturally defined as a joint distribution of the triple of vectors (velocity of incidence, velocity of reflection, normal to the surface at the point of impact). There is vast literature in natural sciences dedicated to rough surfaces. A variety of models of real rough surfaces utilizing periodic, fractal, random functions, etc. have been developed. On the contrary, we provide a unique description of all geometrically possible rough surfaces (where the molecular structure of real bodies is ignored).

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There is a huge variety of shapes of roughness, and it seems probable that the variety of the corresponding scattering laws is also very large. A chapter of the book is dedicated to characterization of scattering laws. In very general terms the solution is the following: a joint distribution of two or three vectors is a law of scattering by a rough surface if, first, it is symmetric with respect to a certain vector exchange, and second, two natural projections of this distribution coincide with some predetermined measures.

We believe that studying billiard scattering by rough surfaces is of potential interest for space aerodynamics. Consider again an illustrative example of a spaceship moving through an interstellar cloud. Imagine that as a result of movement of astronauts in inner compartments the ship very slowly turns around its center of mass in a random uncontrollable fashion, that is, somersaults. Originally the ship is a convex body. Our goal is to apply a roughening on its surface so that the (time averaged) resulting resistance force is minimal. This problem reduces to minimizing a certain functional defined on the set of scattering laws and can be reformulated in terms of optimal mass transportation, where the initial and final mass distributions are concentrated on the unit sphere and correspond to the distributions over velocities of the incident and reflected particle flows. The mass transfer is identified with the scattering law, and the cost of the transfer with the resistance force. A separate chapter is devoted to solving special problems of mass transportation related to the problems of minimal resistance we are interested in.

We will see that the force of resistance of a slowly somersaulting body can be decreased by means of roughening by 3.05% at most. The very fact that the resistance can be decreased by roughening is quite surprising and contradicts the intuition; on the other hand, insignificance of the decrease is disappointing. (Notice that a ’wrong’ roughening can result in an (at most twofold) increase of the resistance — this fact does not look strange at all.) In the case of fast rotation the relation between the roughness and the body dynamics is much more complicated and diverse; we study here the simplest example of a spinning rough two-dimensional disc.

In chapter 1 the basic mathematical notions which are then used throughout the book are defined, and a brief review of the main results is given. Our intention is that the reader who reads only this chapter should get a clear idea on the main results of the book (but not on their proofs). In chapter 2 problems of minimal resistance as applied to translational motion of bodies in a medium are considered. In chapter 3 a generalization of Newton’s problem to convex axisymmetric bodies moving in media with positive temperature is studied. Auxiliary results on billiard scattering by nonconvex and rough bodies are stated in chapter 4. In chapter 5 some special problems on optimal mass transportation are solved explicitly. We believe they are of independent interest, since they extend the (quite short at present) list of explicitly solvable optimal transportation problems. The results of chapters 4 and 5 are used in the next chapter 6, where the problems of minimum and maximum resistance for translating and at the same time slowly rotating (somersaulting) bodies are considered. In chapter 7 the Magnus effect is studied. This effect means that there exists a nonzero transversal component of the force

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acting on a spinning body in a flow of particles. In chapters 8 and 9 billiards possessing extremal properties of best and worst streamlining are studied. Namely, we design bodies of zero resistance and bodies invisible in one and two directions, on one hand, and bodies reversing the direction of particle flows, on the other hand.

I am grateful to G. Buttazzo, A. Stepin and E. Lakshtanov for fruitful discussions on the subject. It was G. Buttazzo who persistently persuaded me to write this book. The work originated from reading the book by V. Tikhomirov ’Stories about maxima and minima’ when preparing my classes for undergraduate students. Many results of the book are co-authored with P. Bachurin, P. Gouveia, K. Khanin, J. Marklof, G. Mishuris, V. Roshchina, T. Tchemisova and D. Torres. Some results are based on personal communi-cations by V. Protasov and J. Zilinskas. I am very grateful to all of them. Last but not least, I want to thank my wife Alla for her patience and continued support of my work.

The work has been partly supported by FEDER funds through COMPETE– Operational Programme Factors of Competitiveness and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690; by the FCT research projects PTDC/MAT/72840/2006 and PTDC/MAT/113470/2009; and by the Grants of President of Russia for Leading Scientific Schools NSh-8508.2010.1 and NSh-5998.2012.1.

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Chapter 1 Notation and synopsis of main

results

In this chapter we introduce the main mathematical notation that will be used throughout the book and state the main results of the book. The proofs of these results are given in the next chapters 2 – 9.

1.1 Definition of resistance

Consider Euclidean space Rd_{, d}_{≥ 2.}

Definition 1.1. A bounded subset of Rd_{with piecewise smooth boundary is called a body}

and is denoted by B. As usual, a convex body is a convex set with non-empty interior. Throughout what follows, convex bodies are assumed to be bounded and are denoted by C.

Remark 1.1. According to this definition, but contrary to physical intuition, a body is not necessarily connected. This is because we do not require this in most of the results presented in the book. When we nevertheless need the condition, we speak of a ’connected body’.

Remark 1.2. In sections 2.5 and 9.1.1 we consider unbounded sets with piecewise smooth boundary; in this case we use the term unbounded body.

Note that a convex body does not necessarily have a piecewise smooth boundary, so a convex body is not necessarily a ’body’.

For a regular point ξ _{∈ ∂C we denote the unit outward normal to ∂C at ξ by n(ξ) and} supply ∂C× Sd−1 _{with the measure µ = µ}

∂C by the formula dµ(ξ, v) = bd|n(ξ) · v| dξ dv,

where dot means the inner product and dξ and dv are the (d_{− 1)-dimensional Lebesgue} measures on ∂C and Sd−1_{, respectively. The quantity b}

d= Γ(d+1₂ )π(1−d)/2is a normalizing

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coefficient chosen so that µ ∂C_{× S}d−1_{= 2|∂C|. It is the reciprocal of the volume of}

the unit (d_{− 1)-dimensional ball; in particular, b}2 = 1/2 and b3 = 1/π. We consider the

measurable spaces

(∂C _{× S}d−1)_±:=_{{(ξ, v) ∈ ∂C × S}d−1:_{±n(ξ) · v ≥ 0}} with induced measure µ. Informally speaking, (∂C× Sd−1₎

− and (∂C× Sd−1)+ are sets of

particles coming into C and going out of C, respectively, and µ measures the (normalized) number of incoming or outgoing particles. We have µ (∂C_{× S}d−1₎

± = |∂C|, so that the

number of particles incoming across ∂C, as well as the number of outgoing particles, is equal to the surface area of C.

In the sequel we will also use the notation

(∂C × A)± :={(ξ, v) ∈ ∂C × A : ±n(ξ) · v ≥ 0},

where A is a subset of Rd_.

The involutive map _{I = I}C : (ξ, v)7→ (ξ, −v) is defined on ∂C × Sd−1, and it maps

(∂C _{× S}d−1₎

− one-to-one onto (∂C × Sd−1)+ (and vice versa).

Consider a body B _{⊂ C and the billiard in R}d_{\ B. We define a map T}

B,C : (ξ, v)7→

(ξ+_B,C(ξ, v), v+_B,C(ξ, v)) between the subspaces (∂C_{× S}d−1₎

− and (∂C× Sd−1)+ as follows.

Let (ξ, v)_{∈ (∂C × S}d−1₎

−. A billiard particle starts its motion from the point ξ with

velocity v, moves in C\ B for some time, possibly reflecting from the boundary of B (this may not happen), and finally crosses ∂C again, at a point ξ_B,C+ (ξ, v) and with velocity v_B,C+ (ξ, v), and leaves C (see Fig. 1.1). In particular, if ξ happens to be a regular point of ∂B, then the period of time when the particle stays in C reduces to a point, in this case we set ξ_B,C+ (ξ, v) = ξ and v_B,C+ (ξ, v) = v_{− 2(n(ξ) · v) n(ξ).}

C

B

b b v ξ v+ ξ+ n(ξ) Figure 1.1: Billiard in Rd_{\ B.}

The map TB,C thus defined establishes a one-to-one correspondence between

full-measure subsets of the spaces (∂C_{× S}d−1₎

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µ and satisfies the equality T_B,C−1 = _{I T}B,CI. In fact, this map determines the billiard

scattering in Rd_{\ B.}

Notice that v_B,C+ can be extended to a function v_B+on a full-measure subset of Rd_×Sd−1

which specifies the velocity of the reflected particle whose position and velocity at an arbitrary moment t before being reflected are equal to ξ + vt and v, respectively. We point out that the function v+_B is translation invariant: v_B+(ξ + vτ, v) = v_B+(ξ, v) for real τ .

We shall consider functionals of the form Rχ[TB,C] =

Z

(∂C×Sd−1₎₋

c(v, v_B+(ξ, v))· |n(ξ) · v| dξ dχ(v), where χ is a Borel probability measure on Sd−1 _and

c : Sd−1× Sd−1 _{→ R}q_, _q_{≥ 1}

is a (generally vector-valued) continuous function satisfying the condition

c(v, v) = 0. (1.1)

Thus, the functional R[TB,C] also takes values in Rq.

Proposition 1.1. If B_{⊂ C}1 and B ⊂ C2, then Rχ[TB,C1] = Rχ[TB,C2].

Proof. Let (∂C × Sd−1₎B

− be the set of values (ξ, v) ∈ (∂C × Sd−1)− such that the

corre-sponding billiard particle reflects from ∂B at least once, and let Tess

B,C be the restriction of

the map TB,C to (∂C× Sd−1)B₋. The restriction of this map to the complementary subset

preserves the second component v, that is, v_B+(ξ, v) = v.

For each (ξ, v) _{∈ (∂C}1 × Sd−1)B₋ the line ξ + vt, t ∈ R has a non-empty intersection

(namely, one or two points) with ∂C2. Let ξ′ be a point in this intersection such that

(ξ′_{, v)}_{∈ (∂C}

2× Sd−1)−, and let TC1,C2,B(ξ, v) := (ξ′, v). The map

TC1,C2,B : (∂C1× Sd−1)

B

− → (∂C2× Sd−1)B₋

thus defined is one-to-one and leaves invariant the second component v. Moreover, it satisfies the relation _T_C−1₁_,C₂_,B =_TC2,C1,B and preserves the measure |n(ξ) · v| dξ dχ(v) for

any such χ. Finally,

T_B,Cess₁ =_{I T}C2,C1,BI T

ess

B,C2TC1,C2,B. (1.2)

Since c(v, v) = 0, it follows that Rχ[TB,C1] =

Z

(∂C1×Sd−1)B−

c(v, v_B+(ξ, v))_{|n(ξ) · v| dξ dχ(v).}

We make the change of variables (ξ, v)7→ (˜ξ, v) =TC1,C2,B(ξ, v) in this integral. By (1.2),

T_B,Cess₁(ξ, v) =_ITC2,C1,BIT

ess

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and we have v_B+(ξ, v) = v_B+( ˜ξ, v) since the second component v is unchanged under ITC2,C1,BI. Furthermore, |n(ξ) · v| dξ dχ(v) = |n(˜ξ)· v| d˜ξ dχ(v). Thus, Rχ[TB,C1] = Z (∂C2×Sd−1)B₋ c(v, v_B+( ˜ξ, v))|n(˜ξ)· v| d˜ξ dχ(v), so that Rχ[TB,C1] = Rχ[TB,C2]. The proof of Proposition 1.1 is complete.

This proposition shows that the value of Rχ[TB,C] depends only on B and not on the

choice of the ambient convex body C. Hence we may write Rχ(B) in place of Rχ[TB,C],

Rχ(B) :=

Z

(∂C×Sd−1₎₋

c(v, v+_B(ξ, v))_{|n(ξ) · v| dξ dχ(v).} (1.3) The functional R is interpreted as the force of resistance of the medium acting on the body, where the distribution of the particles over velocities (in a reference system connected with the body) is given by χ. The concrete value of the integrand c is defined by the concrete mechanical model serving as a prototype for the problem under consideration. So, the function c(v, v+_{) = v}_−v+_{corresponds to the case where a flow of particles falls on}

a resting body, besides the distribution of velocities in the flow is given by χ. In this case Rχ in (1.3) is the force of resistance of the body to the flow. The integrand v− vB+(ξ, v)

is proportional to the momentum transmitted to the body by an individual particle. The function c(v, v+_{) = (v}_−v+₎_{·v corresponds to the case of a parallel flow of particles}

impinging on the resting body, with the direction of the flow being a random variable on Sd−1_{with distribution χ. In this case the value R}

χ is the expectation (mean value) of the

component of pressure force of the flow along the direction of the flow. The integrand (v − v+

B(ξ, v))· v is proportional to the projection of the momentum transmitted to the

body by an individual particle on the direction of the flow.

In some other settings of mechanical problems one has to take other functions c (both scalar and vector-valued). Some of these functions are considered in chapter 7 dedicated to problems of resistance optimization for rapidly rotating rough bodies.

1.2 Newton’s aerodynamic problem

Here we describe Newton’s aerodynamic problem (or problem of minimal resistance) and its generalizations and state some new results obtained in this area in 1990s and 2000s.

We consider the three-dimensional case, d = 3. Let c(v, v+_{) = (v}_{− v}+₎_{· v and let δ} v0

be the probability measure on S2 _{concentrated at a point v}

0 ∈ S2. The functional Rδ_v0

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motion of a body with velocity_−v0 (or, which is the same, the longitudinal component of

the pressure force of a parallel flow of particles at the velocity v0 impinging on the resting

body).

Consider a right circular cylinder Ch of height h and unit radius and a unit vector v0

parallel to the axis of the cylinder. The problem is to find the minimum value of Rδ_v0(B)

in the class of convex bodies B lying in Ch and tangent to all its elements.

We can represent Rδ_v0 in a convenient analytic form. Take an orthonormal system of

coordinates x1, x2, x3 such that the cylinder has the form Ch ={(x1, x2, x3) : x21 + x22 ≤

1, −h ≤ x3 ≤ 0}, and v0 = (0, 0,−1). The upper half of the surface of B is the graph

of a function −fB, where the opposite function fB : Ball1(0) → [0, h] is convex and

Ball1(0) ⊂ R2 is the unit ball x21 + x22 ≤ 1. In view of (1.3), the functional Rδ_v0 takes the

form

Rδ_v0(B) =

Z

Ball1(0)×{0}

(v0− vB+(ξ, v0))· v0dξ, (1.4)

where ξ = (x1, x2, 0) and dξ is two-dimensional Lebesgue measure. Considering that each

particle impinging on the body hits it precisely once, so that v+_B(ξ, v0) = v0+ 2(1 +|∇fB(x1, x2)|2)−1 − ∂fB ∂x1 (x1, x2), − ∂fB ∂x2 (x1, x2), 1 , we see that Rδ_v0(B) = 2R(fB), where

R(f) = Z Z Ball1(0) dx1dx2 1 +|∇f(x1, x2)|2 . (1.5)

Thus, the problem of minimum resistance takes the following form.

Problem 1. Find infR(f) in the class of convex functions f : Ball1(0) → [0, h].

Initially, the problem of minimum resistance was considered by Newton [45] for a narrower class of convex bodies B, which do not merely lie in the cylinder Ch and touch

its lateral surface, but also are symmetric relative to the vertical axis Ox3. In that case

the function fB describing the upper half of the surface of B is radial: fB(x1, x2) = ϕB(r),

where r =px2

1+ x22, and the problem takes the following form.

Problem 2. Find inf Z 1 0 r dr 1 + ϕ′ 2_(r) (1.6)

in the class of convex non-decreasing functions ϕ : [0, 1]_{→ [0, h].}

The solution of Problem 2 (which Newton presented in geometric form and without proof) has the following form in the modern notation:

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and for r0 ≤ r ≤ 1 the function ϕ is described parametrically by r = r0 4 (u 3_{+ 2u +} 1 u) ϕ(r) = r0 4 3 4u 4_{+ u}2_{− ln u −} 7 4 , 1≤ u ≤ u0.

Here r0 = r0(h) and u0 = u0(h) can be found from the system of equations

r0 4 u3₀+ 2u0+ 1 u0 = 1, r0 4 3 4u 4 0+ u20− ln u0− 7 4 = h, u0 ≥ 1.

A brief exposition and an elementary (accessible to high-school children) solution of New-ton’s problem can be found in Tikhomirov’s paper [76], as well as in his book [77].

The solution of Newton’s problem is a body bounded from above and from below by flat discs and reminds a truncated cone with slightly inflated lateral surface. Figure 1.3(a) gives a good idea of its shape for h = 2. Notice that the lateral surface forms the angle 1350 _{with the front surface (upper disc) along the disc boundary.}

Problem 1 has been intensively studied since the early 1990s (see [9, 10, 13, 14],[18]-[20],[34]-[36]). It is known to be soluble, and the solution does not coincide with Newton’s radial solution. It was found numerically in [34], however the properties of the solution are not well understood until now. In addition, the solution of the problem inf_{f ∈D(h)}R(f) in a narrower class D(h) was found analytically in [35]. Functions g in this class have the form gK = fB(K), where−fB(K) describes the upper half of the surface of the set

B(K) = Conv[(Ball1(0)× {−h}) ∪ (K × {0})].

and K ⊂ Ball1(0) is an arbitrary two-dimensional convex set. Here and in what follows,

Conv denotes the convex hull. Thus, B(K) is the convex hull of the union of the circular base Ball1(0)×{−h} and the convex set K ×{0} contained in the horizontal plane Ox1x2.

Notice that D(h) contains the class of convex radially symmetric functions from Ball1(0)

to [−h, 0].

We depict the solution of this problem for h = 2 in Fig. 1.2, where the set K is a horizontal interval with midpoint at the origin.

Some results in the problem of least resistance have also been obtained for nonconvex bodies under the condition that each particle hits the body at most once (this assumption about the shape of the body is called the single impact assumption); see [14],[18]-[20].

Further in this book we consider problems on optimization of resistance in various classes of bodies, mostly nonconvex. In general, particles collide with a nonconvex body several times, so one cannot use simple analytic formulae like (1.5) or (1.6) to calculate the resistance. Instead of this we have to study billiards in the exterior of the body; besides, in several cases the optimization problems are reduced to special problems of optimal mass transfer.

In the next sections 1.3–1.9 synopsis of the main results of the book is given; to each chapter corresponds a separate section.

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Figure 1.2: A convex non-axisymmetric body in the class D(h), h = 2 having minimum resistance.

1.3 Problems of least resistance to translational

motion of nonconvex bodies

If χ = δv0, we have a parallel flow of particles at the velocity v0 falling on a resting body.

The value of the corresponding functional Rδ_v0(B) in an appropriate reference frame has

the form

Rδ_v0(B) =

Z

R2_×{0}

c(v0, vB+(ξ, v0)) dξ. (1.7)

We choose a reference frame such that v0 = (0, 0,−1) and the body B lies in the

half-space x3 ≤ 0. In fact, the body lies in a sufficiently large cylinder Ballr(0)× [−H, 0], and

in (1.7) we integrate over the top base of the cylinder Ballr(0)× {0}, while outside the

base we have v_B+(ξ, v0) = v0, so the integrand vanishes.

The function c is continuous and non-negative and satisfies c(v, v) = 0. In the special case of c(v, v+_{) = (v}_−v+₎_{·v the integral (1.7) has a straightforward physical interpretation:}

this is the resistance produced by the medium to the translational motion of a body with velocity _−v0.

Note that although the function v_B+ is measurable on a full-measure subset of R3_{× S}2_,

its restriction to the subspace v = v0 of measure zero is not necessarily defined. So we

assume in addition that the restriction of v+_B to the subspace v = v0 is a function defined

almost everywhere and measurable with respect to Lebesgue measure in R3 _{× {v} 0}. In

effect this means that the scattering of particles falling in the direction of v0 is regular.

We assume that this condition holds for all bodies considered throughout this section. In chapter 2 we consider a generalized Newton problem of the body of least resistance; generalized because we are looking for the minimum in wider classes _{P(h) and S(h) of} nonconvex bodies inscribed in a fixed cylinder.

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Let_{P(h) be the class of connected (nonconvex in general) sets B lying in the cylinder} Ch = Ball1(0)× [−h, 0] and such that the orthogonal projection of B on the plane Ox1x2

is the disc Ball1(0). This is a broader class than the ones discussed in section 1.2 above.

The problem of minimum resistance in this class has an unexpected answer: inf

B∈P(h)Rδv0(B) = 0. (1.8)

That is, the resistance of bodies inscribed in a fixed cylinder can be made arbitrarily small.

We also consider the class _{S(h) of connected sets lying in the cylinder C}h and

con-taining at least one section Ball1(0)× {c}, −h ≤ c ≤ 0 of it. This is a subclass of the

previous class, _{S(h) ⊂ P(h), but nevertheless it is broader than the classes considered} before. The answer in this class is the same:

inf

B∈S(h)Rδv0(B) = 0. (1.9)

Next we consider a cylinder with arbitrary (not necessarily circular) base and show that the infimum of resistance of bodies inscribed in this cylinder is also equal to zero.

Further, we consider the class of connected bodies B such that C1 ⊂ B ⊂ C2, where C1

and C2 are fixed bounded connected bodies in R3 such that C1 ⊂ C2 and ∂C1∩ ∂C2 =∅.

Again,

inf

C1⊂B⊂C2

Rδ_v0(B) = 0. (1.10)

The relation (1.10) can be interpreted as follows. Any convex body can be transformed within the ε-neighborhood of its boundary so that when the resulting body moves in the prescribed direction in a medium of resting particles, it encounters a resistance smaller than an arbitrarily prescribed quantity ε > 0.

Then we consider the minimization problem for analogues of these classes in the two-dimensional case, d = 2. In this case the least resistance is always positive. We find it explicitly for c(v, v+_{) = (v}_{− v}+₎_{· v.}

Finally, we consider the problem of least specific resistance for unbounded bodies. This problem was first stated by M. Comte and T. Lachand-Robert in [20] under the single impact assumption. We do not impose this assumption; so, in our setting a particle may collide several times with the body surface. We find, in particular, that the infimum of the specific resistance of bodies containing a fixed half-space in a flow perpendicular to the boundary of the half-space equals one half of the resistance of the half-space itself.

1.4 Generalized Newton’s problem in media with

positive temperature

In chapter 3 we address the problem of minimum resistance to translational motion of bodies in a medium with thermal motion of particles. This problem, like the classical

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Newton problem, is considered in the class of convex axisymmetric bodies with fixed length and width.

While the solution of this problem is conventional, the solutions are more diverse. Unlike in the original Newton problem, one has to take into account the composition of the medium: the solution for a homogeneous (and therefore, containing molecules of the same mass) gas is not the same as for a gas consisting of several homogeneous components (and so, containing molecules of different masses).

In the three-dimensional case there are two distinct kinds of solutions. A solution of the first kind is similar to the solution of the classical Newton problem, that is, its surface can be described in the same way as the surface of Newton’s solution. Notice that, unlike in the Newton solution, the angle between the lateral surface and the front disc at the points of disc boundary is not generally 1350_.

A solution of the second kind is a union of two bodies similar to Newton’s solution ’glued together’ along the rear parts of their surfaces. The length (along the direction of the motion) of the front body is always larger than that of the rear ’reversed’ body.

Letting h and the velocity distribution of the particles fixed and changing the velocity V of the body, we find that a solution of the first kind is realized for V _{≥ V}c, and of

the second kind for V < Vc, where Vc = Vc(h) is a certain critical value depending on h.

We present examples of solutions of the first and second kinds in Fig. 1.3(a) and in Fig. 1.3(b), respectively. In these figures and in what follows the body is assumed to move upwards.

In the two-dimensional case, d = 2, the classification of solutions is somewhat more complicated. There exist 5 kinds of solutions (see Fig. 1.4(a) – 1.4(e)):

(a) a trapezium;

(b) an isosceles triangle;

(c) the union of a triangle and a trapezium; (d) the union of two isosceles triangles;

(e) the union of two triangles and a trapezium.

Solutions (a) – (d) are realized for arbitrary velocity distributions of the particles and for arbitrary V ; solution (e) is realized only for some of them. The optimal shapes (a) – (d) appear in the simplest case of homogeneous monatomic gas, while shape (e) can appear in the case where the gas is a mixture of at least two homogeneous components. The numerical computation of the solution (e) is a hard task, which is as yet unsolved. We note that in the two-dimensional analogue of Newton’s problem (that is, with zero temperature) there are only two optimal shapes corresponding to the cases (a) and (b).

In the limit cases, when the speed of the body is large or small in comparison with the mean speed of the particles, the shape of the body of least resistance is universal: it depends only on the length h but does not depend on the velocity distribution of the particles. In the first limit case (V _{→ +∞) the optimal body coincides with the solution} of the classical Newton problem. In the second limit case (V _{→ 0), for d = 3, the optimal}

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(a) V = 1, h = 2 (b) V = 1, h = 3

Figure 1.3: Solutions of the three-dimensional problem in the class of convex bodies of revolution of height h whose maximal cross section is a unit circle. The motion proceeds in a rarefied homogeneous monatomic ideal gas; the velocity of the body is V . The mean square velocity of the gas molecules is 1.

body is a second-kind solution symmetric with respect to a plane perpendicular to the direction of motion, and the angle between this plane and the lateral surface at its upper and lower points is always 51.80_{; while for d = 2 the optimal body is one of the four}

figures:

(a) a trapezium if 0 < h < 1.272; (b) an isosceles triangle if h = 1.272;

(c) the union of an isosceles triangle and a trapezium if 1.272 < h < 2.544; (d) a rhombus if h_{≥ 2.544.}

In cases (a) – (c) the inclination of the lateral sides of these figures to the base is 51.80_,

and in case (d) it is larger.

In a homogeneous monatomic ideal gas the velocities of the molecules are distributed in accordance with the Gaussian law. Assume that the mean square velocity of the molecules is 1; then the type of the solution is determined by two parameters: the velocity V of the body and its length h. We define numerically the regions in the parameter plane corresponding to different kinds of solutions; in addition, for some values of the parameters we determine the shape of the optimal body and calculate the corresponding resistance. We carry out this work separately in the two- and three-dimensional cases.

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(a) h = 0.7 (b) h = 3 (c) h = 6 (d) h = 7.83 (e)

Figure 1.4: The two-dimensional problem. The solu-tions in cases (a)–(d) are numerically calculated for a motion with velocity V = 1 in a gas; the gas parame-ters are as in Fig. 1.3.

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1.5 Scattering in billiards

In the next chapters 4 and 5 we do a preparatory work before passing (in chapters 6 and 7) to the case where bodies perform both translational and rotational motion.

Chapter 4 is devoted to billiard scattering by nonconvex and rough obstacles. First we consider the billiard in the exterior of a two-dimensional connected body B. The law of billiard scattering on B is the probability measure ηB describing the distribution

of the pair (ϕ, ϕ+_{), where ϕ is the incidence angle and ϕ}+ _{is the angle of going away}

of a randomly chosen particle incident on the body (Fig. 4.1). The angles are counted counterclockwise from the normal to ∂(Conv B) and belong to [_{−π/2, π/2].}

The law of scattering on a body admits a convenient representation as follows. We define a sequence of hollows on the boundary of the body (there are 3 hollows in Fig. 4.1); the law of billiard scattering in a hollow is a probability measure defining the joint distribution of (ϕ, ϕ+_{) for a randomly chosen particle going into the hollow. Further, the}

scattering law on the convex part of the boundary of the body is determined by the rule ’the angle of incidence is equal to the angle of reflection’ and is a measure concentrated on the diagonal ϕ+ ₌_{−ϕ. The scattering law η}

B is a weighted sum of the scattering laws

in all the hollows of the body and on the convex part of its boundary.

In an arbitrary dimension we define a rough convex body. The law of scattering on such a body is the joint distribution of a triple of vectors (v, v+_{, n): the initial and}

final velocities and the outer normal at the point of collision, for a randomly chosen particle hitting the body. Thus, a scattering law on a rough body_{B is a (not necessarily} probability) measure ν_B on S_{v}d−1 × Sd−1

{v+_} × S_{n}d−1. It is also convenient to consider the

scattering law at a point on the surface of a rough body; it is the conditional measure ν_B_⌋n=n0 defined on S_{v}d−1× S_{vd−1+_}, where n0 is the outer normal to the body surface at that

point.

In informal terms we can describe a rough body as follows: the surface of a convex body is pocked with microscopic hollows (grooves, cracks, etc), so that macroscopically the resulting (rough) body with hollows looks precisely convex, but billiard scattering on it can be utterly different. The mathematical definition is as follows: a rough body is associated with a sequence of bodies with hollows of size approaching zero. In addition, a sequence of such bodies must satisfy the condition of convergence of the sequence of corresponding scattering laws. Furthermore, an equivalence relation between such sequences is defined, and the convention is that equivalent sequences of bodies represent the same rough body. Otherwise we can say that a rough body is obtained by grooving a fixed convex body. Clearly, a convex body can be grooved in infinitely many ways, differing (informally speaking) by the shape of hollows.

In Theorems 4.1 – 4.5 we give a complete characterization of scattering laws. Each statement has roughly the same form: we assert that a measure is a scattering law if and only if it has fixed marginals and possesses a certain symmetry property.

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As examples we give two such statements (Theorem 4.3 and Corollary 4.2). First, a measure on [_{−π/2, π/2]}2 _{can be weakly approximated by scattering laws η}

B if

and only if it is invariant relative to the exchange of variables (ϕ, ϕ+₎_{7→ (ϕ}+_{, ϕ) and both}

its natural projections on [_{−π/2, π/2] (that is, its marginals) coincide with the measure} λ given by dλ(ϕ) = 1

2 cos ϕ dϕ.

Second, a measure on S_{v}d−1×Sd−1

{v+_} is a scattering law at a point of a rough body, if and

only if it is invariant relative to the transformation (v, v+₎_{7→ (−v}+_,_{−v) and its natural}

projections on S_{v}d−1 and S_{vd−1+_} are probability measures λ_−n and λn with the densities

bd(v· n)− and bd(v+· n)+, respectively. The normalizing coefficient bd is defined in section

1.1, and n is the outward normal to the body surface at the given point. Notice that the measures λ_−n and λn define the distributions of the incident and reflected flows of

particles over velocities.

1.6 Problems of optimal mass transportation

We will see in chapter 6 that some problems of resistance optimization for rough surfaces, with the use of the aforementioned theorems 4.1 – 4.5, can be reduced to a problem of finding the measure with fixed marginals (that is, the scattering law) on [_{−π/2, π/2]}2 _or

(Sd−1₎2 _{minimizing a certain linear functional. This problem in general is as follows.}

Consider measurable spaces (X, λ1) and (Y, λ2) such that λ1(X) = λ2(Y ) and a

con-tinuous function c : X _{× Y → R (usually called cost function). Let Γ(λ}1, λ2) be the set

of measures ν on X _{× Y whose marginals (projections on X and Y ) are, respectively,} λ1 and λ2. (This means that for any two measurable sets A1 ⊂ X and A2 ⊂ Y holds

ν(A1× Y ) = λ1(A1) and ν(X× A2) = λ2(A2).) The problem of minimization

inf

ν∈Γ(λ1,λ2)

ZZ

X×Y

c(x, y) dν(x, y) (1.11) is called the problem of optimal mass transportation, or the Monge-Kantorovich problem. This problem can be interpreted as follows. We have two mass distributions given by the measures λ1 and λ2 on X and Y , respectively, and a function c(x, y) defining the

cost of transfer of a unit mass from x _{∈ X to y ∈ Y . A plan of mass transfer from the} initial position λ1 to the final position λ2 (or just a transport plan) is given by a measure

ν with marginals λ1 and λ2, and the total cost of the transfer with this plan is equal to

the integral in (1.11). One needs to find the optimal transfer plan, that is, the measure ν_∗ minimizing the transfer cost.

In general it seems impossible to provide exact solution for an optimal transportation problem. The known cases of exactly soluble problems are quite rare; they are rather exceptions to the general rule. The case of the one-dimensional transport, where X and Y _{⊂ R, c(x, y) = f(x ± y), and f is strictly convex or concave, is the simplest one; then} the optimal plan is monotone, that is, is given by a measure supported on the graph

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of a monotone function (see, e.g., [42]). We note a very interesting case considered by McCann [42] where c(x, y) = f (_{|x − y|) and f is a positive strictly concave function.} Note also the case where the measures λ1 and λ2 coincide and are uniform on the segment

X = Y = [0, 1], with c(x, y) = h(x+y) or h(x_{−y), where h has 3 intervals of monotonicity} [79].

Below we cite some explicitly soluble cases of the two-dimensional mass transportation problem, where λ1 and λ2 are Lebesgue measures on compact sets X and Y in R2, and

the cost function is the Euclidean distance, c(x, y) =_{|x − y|. The following examples are} taken from the papers by Levin [38, 40, 39].

1. The set Y is obtained by shifting X by a vector b _{∈ R}2_{, that is, Y = X + b. This}

shift actually produces an optimal transportation; in other words, the measure supported on_{{(x, y) : x ∈ X, y = x + b} ⊂ R}4 _{is an optimal transport plan.}

2. X is a rectangle of size 1_{× 2 and Y is the rectangle obtained by rotating X by 90}0

about its center.

3. X is an equilateral triangle and Y is the triangle obtained by rotating X by 600

about its center.

4. X is an equilateral triangle and Y is a triangle obtained by reflecting X relative to one of its sides.

5. X is a square and Y is the square obtained by rotating X by 450 _{about its center.}

In all these cases the optimal transfer is generated by piecewise isometrical transfor-mations.

In chapter 5 some special optimal transfer problems are explicitly solved. First we consider a problem of mass transport from R to R with a cost function of the form c(x, y) = f (x + y), where f is an odd function strictly concave on R+ = {x ≤ 0} (and

therefore strictly convex on R₋ ={x ≤ 0}), in the case where the initial mass distribution coincides with the final one, λ1 = λ2. We impose some additional technical conditions on

λ1.

We show that the optimal measure is uniquely defined by its support, which belongs to the union of two lines on the plane: the ray x = y_{≥ 0 and a curve symmetric relative} to this ray (see Fig. 5.1). The curve belongs to a finite- or countable-parameter family of curves, which does not depend on f and is defined merely by λ1, while the choice of the

optimal curve from this family is defined by f .

In an important particular case the family is one-parameter, and therefore the problem reduces to minimization of a function of a real variable.

Further we consider a special problem of mass transport on the unit sphere in Rd_.

The initial and final spaces are complementary hemispheres, X = S_−nd−1 := {x ∈ Sd−1 _:

x_{· n ≤ 0} and Y = S}d−1

n := {x ∈ Sd−1 : x· n ≥ 0}, and the measures λ1 and λ2 are

defined by the following condition: the orthogonal projection of each measure on the plane x_{· n = 0 coincides with the Lebesgue (d − 1)-dimensional measure on the circle} {x : x · n = 0, |x| ≤ 1}. The cost function is the squared distance, c(x, y) = 1

2 |x − y| 2_.

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This problem can be naturally interpreted in terms of billiard scattering by rough surfaces. Fix a point on a rough surface and let n be the outward normal to the surface at this point. The flows of incident and reflected particles are identified with the hemispheres X and Y , respectively. With this identification, to any incident or reflected particle we assign its velocity v or v+ _{(we have v}_{· n < 0 and v}+ _{· n > 0). The measures λ}

1 and

λ2 describe the densities of the incident and reflected flows. Each admissible measure

η _{∈ Γ(λ}1, λ2) defines a billiard scattering at that point (more precisely, the symmetrized

measure ηsymm = 1₂(η + π_d#η) is a scattering law at a point; here πd exchanges the

ar-guments, πd(x, y) = (y, x), and π#d is the induced map of measures). The cost function 1

2|v − v

+_|2 _{is the (normalized) momentum transmitted to the body by a particle with the}

corresponding velocities of incidence and reflection, and the total cost of the transfer is the specific resistance at the point.

Since this problem possesses the axial symmetry relative to n, one can show that the optimal transfer is performed along the meridians (we take the sphere poles to be n and _{−n) and is axially symmetric. As a result one comes to a one-dimensional problem} identical to the one considered earlier in chapter.

Two schemes of mass transfer along the meridians are depicted in Figures 1.5 (a) and 1.5 (b). The transfer shown in Fig. 1.5 (a) is induced by the law ϕ+ _{= ϕ (’the angle of}

incidence = the angle of reflection’) and corresponds to reflection from a smooth surface. It is instructive to consider an argument showing that it is not optimal. Consider two small arcs I1 and I2 adjoining the equator and reverse the monotonicity of the transfer

from I1 to I2, that is, replace monotone increasing with monotone decreasing. Since these

arcs are ’almost’ rectilinear and the cost function equals the squared distance, the transfer cost will decrease under this reversal.

In Fig. 1.5 (b) the optimal transfer scheme is depicted in the case d = 2. Both upper and lower halves of the meridian are divided into pairs of arcs, the left and the right ones. The transfer between the left arcs is monotone increasing, ϕ+ _{= ϕ, and the transfer}

between the right arcs is monotone decreasing. Notice that the left and right arcs partly overlap; this means that the mass at each point of the ’overlapping zone’ splits in two parts, which are then transported to two different points. In terms of optimal transfer this means that the transfer solves the Monge-Kantorovich problem, but not the Monge one.

In higher dimensions, d ≥ 3, the scheme of optimal transfer is roughly the same; the most significant difference is that the overlapping of the two arcs disappears (or, more precisely, reduces to a point).

1.7 Optimizing the mean resistance

The value of the functional Ru(B) in (1.3) with the cost function c(v, v+) = (v− v+)· v

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n b b b ϕ ϕ+ I1 I2 (a) n b (b)

Figure 1.5: The mass transfer along the meridian (from the lower part of the meridian to the upper one) is marked by arrows. The transfer induced by shift along n is shown in figure (a), and the optimal transfer is shown in figure (b).

a translational motion in Rd _{in a medium of resting particles, with velocity v randomly}

chosen from a uniform distribution in Sd−1_{. The resistance R}

δv(B) to this motion (more

precisely, the projection of the resistance force on the direction of motion) is a random variable, and its mathematical expectation equals Ru(B). Note that the cost function can

be written as c(v, v+_{) =} 1

2|v − v+|2.

We can propose another interpretation of this functional: the body B moves trans-lationally with fixed velocity and at the same time slowly rotates. The rate of rotation is small enough that we can neglect it in interactions of the body with individual parti-cles. In a reference system attached to the body the velocity vector draws a curve on the sphere Sd−1 _{thus inducing a (singular) probability measure on the sphere: the measure of}

a subset of Sd−1 _{is the (normalized) total time when the vector lies in this subset within a}

certain period of observation. We assume that as the period of observation extends, this measure weakly converges to u. Then the mean resistance over this period approaches Ru(B).

The problems of minimizing and maximizing Ru(B) are studied in chapter 6 in different

classes of bodies. The mathematical tools necessary for this study are elaborated in the previous chapters 4 and 5.

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repre-sented in the form

Ru(B) = |∂(ConvB)|

Z Z

(1 + cos(ϕ_{− ϕ}+)) dηB(ϕ, ϕ+),

where = [−π/2, π/2] × [−π/2, π/2]. We state the following problem. Problem 3. Find inf Ru(B)

(a) in the class of connected (generally speaking, nonconvex) bodies B of fixed area; (b) in the class of convex bodies B of fixed area.

Using the results of chapter 4 on characterization of the measures ηB, we reduce

Problem 3(a) to the following special optimal transportation problem: inf

η∈Γ(λ,λ)

Z Z

(1 + cos(ϕ_{− ϕ}+)) dη(ϕ, ϕ+) (where λ is the measure on [−π/2, π/2] given by dλ(ϕ) = 1

2 cos ϕ dϕ), which we then

solve using the results of chapter 5. A minimizing sequence of bodies is constructed that can be identified with a rough disc of prescribed area. The solution in Problem 3(b) is a (standard) disc of the same area, and

(the least resistance in the class of nonconvex bodies)

(the least resistance in the class of convex bodies) = m2 ≈ 0.9878.

Allowing some freedom of speech, one can say that the body of least resistance in the class of convex bodies is a disc, and in the class of nonconvex bodies it is a rough disc, and the resistance of the latter body is approximately 1.22% smaller than that of the former one.

Next we consider the following problem. Let C1 and C2 be bounded convex bodies

such that C1 ⊂ C2 ⊂ R2 and ∂C1 ∩ ∂C2 = ∅. We consider the class of convex bodies B

such that C1 ⊂ B ⊂ C2.

Problem 4. Find (a) infC1⊂B⊂C2Ru(B) and (b) supC1⊂B⊂C2Ru(B).

The solution of Problem 4(a) essentially repeats that of Problem 3(a).

In cases (a) and (b) minimizing and maximizing sequences can be identified with rough bodies obtained by grooving C1 and C2, respectively, and we have

infBRu(B) Ru(C1) = m2 ≈ 0.9878 and sup_BRu(B) Ru(C2) = 1.5.

Next we show that the resistance of a body in a medium with thermal motion of particles is proportional to the resistance in the medium consisting of resting particles, with a coefficient which is larger than 1 and depends only on the nature of motion of the

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particles: the higher the temperature, the larger the coefficient, the larger the resistance. Hence in a medium with positive temperature Problems 3 and 4 have the same solution as before.

In the case of an arbitrary dimension d_{≥ 2 we solve problems of optimizing resistance} in the class of rough bodies obtained by grooving a fixed convex body C.

Problem 5. Find (a) 1

R(C) sup{Ru(B) : B is obtained by grooving C};

(b) _R(C)1 inf{Ru(B) : B is obtained by grooving C}.

These ratios appear to depend only on the dimension d, and not on the particular body C. With the use of the results on characterization of measures generated by rough bodies Problem 5 reduces to the problem of optimal mass transportation on Sd−1 _{considered in}

chapter 5. One finds that sup_BRu(B) R(C) = d + 1 2 and inf_BRu(B) R(C) = md, where, in particular, m2 ≈ 0.9878, m3 ≈ 0.9694 and lim d→∞md= 1 2(1 + Z 1 0 pln z ln(1 − z) dz) ≈ 0.791.

We illustrate these results by the following example. Consider a spherical artificial satellite rotating around the Earth and being decelerated by the thin atmosphere. As-sume that the surface of the satellite is made of materials ensuring that molecules of the atmosphere reflect from it elastically. The twofold problem consists in (a) reducing or (b) increasing the resistance to the motion by appropriate grooving the surface of the satellite. It follows from our results that the resistance can be reduced by at most 3.05% or can be at most doubled.

1.8 Dynamics of a spinning rough disc

In chapter 7 we study the resistance and dynamics of rotating bodies. As opposed to the previous chapter, here we consider the case of fast rotation. This means that the product of the angular velocity and the diameter of the body has the same order of magnitude as its translational velocity.

We limit ourselves to the simplest case where a rough disc rotates around the center and at the same time moves through a rarefied medium on the plane. The center of mass of the disc coincides with its geometric center. In this case a simple scheme of scattering is realized, where an incident particle interacts with the body at the point of collision and then goes away forever. Note that any other shape of a convex body (which is not necessarily rough) and any other location of the center of mass may lead to a more

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complicated scheme, where a particle is reflected from the body several times at two or more points of the boundary.

On the other hand, dynamics of nonconvex bodies seems to be even more complicated. The point is that it is natural to consider the interaction of the particle with the body in a reference system attached to the body, but in such a reference system the motion of a particle becomes curvilinear between consecutive collisions, and therefore is difficult to study.

The main feature of dynamics of a rapidly rotating disc is that the force of resistance is not parallel to the velocity of the body. This phenomenon is well known to the physi-cists and is called Magnus effect. The transversal component of the resistance force can be codirectional to the instant velocity of the body’s front point, or can have opposite directions relative to this point. In these cases one speaks of the proper or inverse Magnus effect, respectively. It is well known to the physicists that in relatively dense gases proper effect takes place, while in rarefied media usually the inverse effect is realized.

The Magnus effect in rarefied media is usually derived from nonelastic interaction of gas particles with the body [8, 31, 82, 83]. On the contrary, in our model this effect is due to multiple reflections of particles from the body. The magnitude and direction of the resistance force, as well as the kind of the effect (proper or inverse) depend on the shape of roughness in a complicated way. In our model both kinds of the effect are realized, but the inverse effect dominates in a sense. For any fixed value of relative angular velocity we represent the force acting on the disc and the moment of this force as functionals depending on the ’shape of roughness’ (Theorem 7.1).

The set of admissible forces is a convex set formed by the resistance forces (R1, R2)

corresponding to all possible roughness shapes. The problem of finding the set of ad-missible forces reduces to a vector-valued problem of optimal mass transfer and is then numerically solved for some values of angular velocity (Figures 7.4 and 7.9). Each of these sets is divided by the vertical line R1 = 0 into two unequal parts; the greater part

corresponds to the inverse Magnus effect, and the smaller part, to the proper one. In some simple cases the disc trajectory is found explicitly (see Figure 7.8). In par-ticular, as shows numerical simulation, a single disc with roughness formed by equilateral triangles can demonstrate three different kinds of behavior, depending only on the initial data. If the initial angular velocity is sufficiently small, the disc trajectory is a curve approaching a straight line. If it is sufficiently large, the trajectory is a converging spi-ral, and if it takes an intermediate value, the trajectory coincides with or approaches a circumference.

The following problem remains unsolved: find all curves which can be drawn by the center of mass of a spinning rough disc (or, more generally, an arbitrary body) that moves in a rarefied medium.

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1.9 Billiards possessing extremal aerodynamic

prop-erties

In the last chapters 8 and 9 we study bodies that have the best and the worst aerodynamic properties.

In chapter 8 we concentrate on invisibility and related problems of perfectly stream-lined bodies. The interest to invisibility (creation of ’invisibility coats’, etc) has drastically grown up due to recent developments in metamaterials with unusual optical properties. On the contrary, we examine the effect of invisibility achieved by using only mirror sur-faces.

First we define three classes of bodies: bodies having zero resistance when moving in a fixed direction, bodies leaving no trace when moving in one direction, and bodies invisible in one direction. We say, in particular, that a body is invisible in one direction, if a particle falling on it along a certain straight line in this direction, after making several reflections will eventually move along the same line. We show that these three classes are nonempty, do not coincide, and are embedded one into another.

The very fact of existence of bodies having zero aerodynamic resistance when moving in a medium is surprising. We provide explicit constructions of such bodies (they are depicted in Figures 8.1 – 8.7). In Fig. 8.7, for instance, it is shown how to get an invisible body by making a hole inside a cylinder along its axis.

Notice that a body of zero resistance is supposed to move uniformly in a medium with zero temperature and constant density. If, say, a spaceship having zero resistance turns its engines on and makes a maneuver, the medium will produce a force resisting to maneuvering. Further, when flying into a zone with larger density the ship experiences a decelerating force, and when going out of this zone it experiences a compensatory accelerating force.

Next we design a body invisible in two mutually orthogonal directions (Figures 8.12 and 8.13) and a body invisible from one point (Fig. 8.17). It is impossible, however, to design a body invisible in all directions (Theorem 8.4). There still remain many unsolved questions, first of all: how many directions and/or points of invisibility can be realized?

In chapter 9 we study bodies with the worst aerodynamics properties: retroreflectors. A retroreflector is an optical device that reverses the direction of any incident beam of light. Note that a perfect retroreflector using refraction of light rays is well known in optics: it is the Eaton lens, a transparent ball with refractive index growing from 1 on the ball boundary to infinity at its center. It is unknown, however, if there exist perfect retroreflectors that use only reflection of light rays, that is, billiard retroreflectors. Instead we construct in two dimensions several asymptotically perfect billiard retroreflectors, that is, families of bodies whose reflective properties approach the property of retro-reflection.

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The reflective properties of connected two-dimensional bodes are derived from proper-ties of hollows on their boundary; therefore we concentrate on constructing hollows. Three families of asymptotically retroreflecting hollows are constructed: Mushroom (Fig. 4.7), Tube (Fig. 9.6), and Notched Angle (Fig. 9.9). The first of them admits generalization to higher dimensions. The fourth hollow considered in chapter 9 is called Helmet (Fig. 9.12); it possesses very good properties of retro-reflection, yet it is not perfect.

The four resulting bodies: two-dimensional retroreflectors with the corresponding hol-lows on their boundary — are depicted in Fig. 9.14.

Each of the proposed shapes has its own drawbacks. The number of reflections in Tube and Notched Angle is very large and goes to infinity when the reflective properties of the corresponding shape approach retro-reflection. On the contrary, most particles make only one reflection in Mushroom; however, there always exist a nonzero difference between the directions of incidence and reflection. In addition, as noted by V. Protasov1_{, in practice}

it is impossible to produce a good quality retroreflector with mushroom-shaped hollows, since the size of smallest hollows should be much smaller than the size of atoms (the corresponding estimates are given in Appendix 9.7.3). Helmet seems to be the best in practical applications, especially for the purpose of recognition of the body contour.

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Chapter 2 Problem of minimum resistance to

translational motion of bodies

Newton’s aerodynamic problem consists in minimizing the resistance to the translational motion of a three-dimensional body moving in a homogeneous medium of resting particles. The particles do not interact between themselves and reflect off elastically in collisions with the body. This problem has been considered for various classes of admissible bod-ies. In Newton’s initial setting [45] the class of admissible bodies consisted of convex axisymmetric bodies of fixed length and width, that is, bodies inscribed in a fixed right circular cylinder. The problem was later considered for various classes of (convex and ax-isymmetric) bodies, for example, for bodies whose front generator has a fixed length (and whose width is also fixed) [37, 5], for bodies of fixed volume [6] and so on. A major step forward was made in the 1990s, when unexpected and striking results were obtained for some classes of non-axisymmetric bodies, and later for nonconvex bodies [9, 13, 14],[18]-[20],[34]-[36]). However, the authors kept the initial assumption that the body must have fixed length and width, that is, can be inscribed in a fixed right circular cylinder.

A further constraint imposed on all classes of bodies was as follows: a particle cannot hit the body more than once. Here we do not impose this constraint. In section 2.1 we consider two classes of (generally speaking, nonconvex and non-symmetric) bodies inscribed in a circular cylinder. These classes differ in accordance with the meaning one puts in the expression ’inscribed in a cylinder’. We show that in each class the infimum of the resistance is zero, that is, there exist ’almost perfectly streamlined’ bodies. In section 2.2 this result is generalized to right cylinders with arbitrary (not only circular) section. In section 2.3 we demonstrate that any convex body can be transformed in a small neighborhood of its boundary so that the resulting body displays a resistance less than an arbitrary small ε > 0. In fact, any body can be made ’almost perfectly streamlined’ by making microscopic longitudinal ’grooves’ in its surface. In section 2.4 we consider an analogue of Newton’s problem in the two-dimensional case. Here the minimum resistance is always positive, but smaller than that of convex bodies. In section 2.5 we consider the

Exterior billiards: systems with impacts outside bounded domains

systems with impacts outside bounded domains

Alexander Plakhov

Department of Mathematics, University of Aveiro, Portugal

and

Abstract

Contents

Preface

Chapter 1

Notation and synopsis of main

results

1.1

Definition of resistance

C

B

1.2

Newton’s aerodynamic problem

1.3

Problems of least resistance to translational

motion of nonconvex bodies

1.4

Generalized Newton’s problem in media with

positive temperature

1.5

Scattering in billiards

1.6

Problems of optimal mass transportation

1.7

Optimizing the mean resistance

1.8

Dynamics of a spinning rough disc

1.9

Billiards possessing extremal aerodynamic

prop-erties

Chapter 2

Problem of minimum resistance to

translational motion of bodies